The Witten complex for singular spaces of dimension two with cone-like singularities

摘要

The Witten deformation is a method proposed by Witten which, given a function f : M → R on a smooth compact manifold M, allows to prove the Morse inequalities. Witten's proof of the Morse inequalities is analytical and can thus be applied to situations where the Thom-Smale method is not accessible. In these notes we generalise the Witten deformation to certain singular Riemannian manifolds X which are metric models for singular algebraic curves, and functions on X which we call admissible Morse functions. They are particular examples of stratified Morse functions in the sense of the theory developed by Goresky and MacPherson.